Basics of Algebra: Part II

This series of lessons is designed
to help you learn, or review, the fundamentals of algebra. In this
lesson we continue our discussion of relations and extend this to
solving equations and inequalities.

One of the most useful aspects of
algebra is every day life is using algebraic expressions to model
real life situations. This involves equations and their
solutions.

We begin by reviewing equations:

Equationsare equivalence relations between 2 numbers,
variables or expressions, for example . As we've already
seen, equations such as can be simplified to
their simplest form, , in which we can identify the number our variables
represent. In this case, our number x represents 2.

Note that this doesn't work with simple expressions. If we just
have the expression , there's nothing we
can say about x. It's the "=" symbol that allows us to say
something about x.

Of course, we can't say something
about x all the time. Some equations have no
solution, meaning no real number value for x satisfies the
relation. is a good
example, since when simplifying we get , which is clearly never
true. On the other hand, some equations are true for all
values of x. For example, , which
simplifies to , is always true regardless of x.

Solving equations is easy, but what about
inequalities?

Inequalities, <,>,etc., work
the same way as equalities in terms of simplification, with one
important difference. When we multiply or divide both sides by a
negative value in inequivalence relations, we have to flip our
inequality sign.

For example, say . Then while simplifying we subtract 4 from each
side, and
finally divide by 3, flipping the sign: . Plugging this
into the original inequality shows how important flipping the sign
is- plug in -2, which satisfies the solution, to get , which
is true. If you forget to flip the sign you get , and plugging in
2 for example will result in , which is obviously
wrong.

This is fun! Can we kick it up a notch?

Sure, if you insist, let's get to a
more challenging level. Until now we've dealt with linear equations
with one variable- i.e. we had only one variable and that variables
only had degree 1, so we had no . That we'll do next time,
but for now, let's throw in another variables.

Equations in two
variableslook a lot like
equations with 1 variable. Say . Can you solve for x
and y? No, you can't. There are too many possibilities- and works, but so do
and
.
There are actually infinitely many possible combinations. In
reality, this equation represents a line in the XY plane, but
that's an analytic geometry thing, we're doing algebra.

In order to solve equations in 2 variables, we need at least 2
equations, as Elmo here clearly understands. This is analogous to
finding the intersection of 2 lines in a plane. The same principle
works for higher numbers- you have 5 unknowns? You need 5
equations. So, for our purposes, let's say and . A simple guess
and check shows and . But there are more formal ways of solving equations
in two unknowns.

Substitution:Writing
one variable in terms of the other and substituting it into the
other equation. In our example, we can rewrite equation 2 as
, and
substitute this into equation 1: , and solve from
there.

Elimination:Multiplying the equations by constants so that we
can add/subtract them from each other and eliminate one of the
variables. For example, if we add equations 1 and 2 together we get
,
and we can solve from there.

Is it really that
easy?%

Yes and no. No, because I can throw equations at you with really
ugly irrationals, and it'll take you years to solve. But also Yes,
because that really is all there is to it, the technique is simple.
All you really need now is practice.

Working with more than one variable introduces some other concepts,
like factoring and expanding, which we'll talk about next time.
We'll also finally get to quadratic equations and maybe some
exponents.